C₂×#20276890 #27538827

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#20276890

Splittings

Fitting	C₂×C₅₃×#365 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2231

Subgroups

	subgroup	G/H
center	C₂²	#16810912
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#365	C₂
socle	C₂²×C₅₃	#476
prime radical	C₂×#13133239	C₂²
hypercenter	C₂×#365	D₅₃
hypocenter	C₅₃	C₂×#2231
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

526 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 points:

(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(56,59)(57,64)(58,70)(60,73)(61,77)(62,75)(63,79)(65,82)(66,85)(67,84)(68,86)(69,74)(71,81)(72,80)(76,83)(78,87);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(56,58,69,78,62,71,60,61)(57,63,76,66,67,80,65,68)(59,70,74,87,75,81,73,77)(64,79,83,85,84,72,82,86);(56,57,62,67)(58,68,71,66)(59,72)(60,76,69,65)(61,63,78,80)(64,81)(70,84)(73,86)(74,85)(75,79)(77,82)(83,87);(54,55)

Conjugacy classes & character table

878 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 324303);

C2×#20276890 #27538827